Bridge sections designed by this rule differ little from those designed by formulae based directly on Wöhler's experiments. This rule has been revived in America, and appears to be increasingly relied on in bridge-designing. (See _Trans. Am. Soc. C.E._ xli. p. 156.)
The method of J.J. Weyrauch and W. Launhardt, based on an empirical expression for Wöhler's law, has been much used in bridge designing (see _Proc. Inst. C.E._ lxiii. p. 275). Let t be the _statical breaking strength_ of a bar, loaded once gradually up to fracture (t = breaking load divided by original area of section); u the breaking strength of a bar loaded and unloaded an indefinitely great number of times, the stress varying from u to 0 alternately (this is termed the _primitive strength_); and, lastly, let s be the breaking strength of a bar subjected to an indefinitely great number of repetitions of stresses equal and opposite in sign (tension and thrust), so that the stress ranges alternately from s to -s. This is termed the _vibration strength_. Wöhler's and Bauschinger's experiments give values of t, u, and s, for some materials. If a bar is subjected to alternations of stress having the range [Delta] = f_{ max.}-f_{ min.}, then, by Wöhler's law, the bar will ultimately break, if
where F is some unknown function. Launhardt found that, for stresses always of the same kind, F = (t-u)/(t-f_{ max.}) approximately agreed with experiment. For stresses of different kinds Weyrauch found F = (u-s)/(2u-s-f_{ max.}) to be similarly approximate. Now let f_{ max.}/f_{ min.} = [phi], where [phi] is + or - according as the stresses are of the same or opposite signs. Putting the values of F in (1) and solving for f_{ max.}, we get for the breaking stress of a bar subjected to repetition of varying stress,
f_{ max.} = u(1+(t-u)[phi]/u) [Stresses of same sign.] f_{ max.} = u(1+(u-s)[phi]/u) [Stresses of opposite sign.]
The working stress in any case is f_{ max.} divided by a factor of safety. Let that factor be 3. Then Wöhler's results for iron and Bauschinger's for steel give the following equations for tension or thrust:--
Iron, working stress, f = 4.4 (1+½[phi]) Steel, working stress, f = 5.87 (1+½[phi]).
In these equations [phi] is to have its + or - value according to the case considered. For shearing stresses the working stress may have 0.8 of its value for tension. The following table gives values of the working stress calculated by these equations:--